Resistivity logging measures the electrical resistivity of formations surrounding an earth borehole. Resistivity logging is a commonly used technique of formation evaluation. For example, porous formations having high resistivity generally indicate the presence of hydrocarbons, while porous formations having low resistivity are generally water saturated. A logging device is lowered in the well bore on a wireline cable and measurements are taken with the device as the cable is withdrawn. One technique of resistivity logging uses electrodes. For example, button electrodes are used on a pad which is placed in close proximity to the borehole wall. A measured current flows in a circuit that connects a current source to one electrode, through the earth formation to a return electrode and back to the current source in the logging tool.
FIG. 1 illustrates one example of using electrical earth borehole logging for the determination of formation resistivity, RF. In this example, an earth formation 100 is penetrated by a wellbore 200. Electrodes 10 and 12 are in wellbore 200, in proximity of earth formation 100. Electrode 10 is separated a distance from electrode 12. A standoff zone 20 exists between electrodes 10 and 12 and earth formation 100. Drilling fluids typically reside in wellbore 200, including within standoff zone 20. A current I, having a voltage V, is generated by a signal generator 1 and introduced into formation 100 through electrode 10. Electrode 12 is a ground return. Current I penetrates through standoff zone 20 into formation 100. Standoff zone 20 exhibits a standoff impedance ZC between electrodes 10 and 12, owing to the electrical characteristics of the drilling fluids. For example, Oil Based Mud (OBM) has a high resistance, causing a capacitive dielectric effect between electrodes 10 and 12. A current sensor 2 detects the current at electrode 10, producing voltage signal VI. A comparison is made of the voltage V from signal generator 1 and the detected current, as represented by voltage signal VI. A measure of the formation resistivity, RF is determined from this comparison.
For Oil Based Mud (OBM), a high frequency driving source has to be used to excite the formation in order to measure the formation resistivity. The simplified impedance model of the Oil Based Mud (OBM) standoff and the formation is:Z=ZR+jZC≈RF+jZC(CS,ω)  (1)Where Z is the total impedance expressed as a complex number, ZR is the real component of the total impedance, j is the square root of minus one, Zc is the impedance from the standoff, RF is formation resistance, CS is the capacitance of the standoff, and ω is the frequency in radians.
Given ω=2π·10 MHz as a nominal frequency value, the ratio of impedances for the Oil Based Mud (OBM) can range from 20 to 100 or higher, which is
                    20        ≤                              Z            C                                R            F                          ≤        100                            (        2        )            
Direct measurements of the amplitude and phase of Z are obtained by comparing the voltage V from the signal generator and the detected current, as represented by voltage signal VI. These direct measurements are then used to calculate RF. For example, a fully integrated RF IC for measuring amplitude and the phase between two independent input signals can be used to compare voltage V and VI. In this example, the ZC is dominant so that the observed phase is close to 90°. As a result, the resolutions for both amplitude and phase have to be very high to achieve accurate estimation of formation resistivity, RF using this technique. A method for direct measurement is desired that does not have such a high resolution requirement in order to achieve accurate estimation of formation resistivity, RF.